Question 4.4.9.3: Fitting a logistic function to Data The data in Table 11 rep......
Fitting a logistic function to Data
The data in Table 11 represent the amount of yeast biomass in a culture after t hours.
(a) Using a graphing utility, draw a scatter diagram of the data with time as the independent variable.
(b) Using a graphing utility, build a logistic model from the data.
(c)Using a graphing utility, graph the function found in part (b) on the scatter diagram.
(d) What is the predicted carrying capacity of the culture?
(e) Use the function found in part (b) to predict the population of the culture at t = 19 hours.
| Table 11 | |||
| time (hours) | Yeast Biomass | time (hours) | Yeast Biomass |
| 0 | 9.6 | 10 | 513.3 |
| 1 | 18.3 | 11 | 559.7 |
| 2 | 29.0 | 12 | 594.8 |
| 3 | 47.2 | 13 | 629.4 |
| 4 | 71.1 | 14 | 640.8 |
| 5 | 119.1 | 15 | 651.1 |
| 6 | 174.6 | 16 | 655.9 |
| 7 | 257.3 | 17 | 659.6 |
| 8 | 350.7 | 18 | 661.8 |
| 9 | 441.0 | ||
| Source: Tor Carlson (Über Geschwindigkeit und Grösse der Hefevermehrung in Würze, Biochemische Zeitschrift, Bd. 57, pp. 313–334, 1913) | |||
(a) See Figure 54 for a scatter diagram of the data.
(b) A graphing utility fits a logistic growth model of the form y={\frac{c}{1+a e^{-b x}}} by using the LOGISTIC regression option. See Figure 55. The logistic from the data is
y={\frac{663.0}{1+71.6e^{-0.5470x}}}where y is the amount of yeast biomass in the culture and x is the time.
(c) See Figure 56 for the graph of the logistic model.
(d) Based on the logistic growth model found in part (b), the carrying capacity of the culture is 663.
(e) Using the logistic growth model found in part (b), the predicted amount of yeast biomass at t = 19 hours is
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